ABSOLUTELY CONVERGENT FOURIER SERIES AND GENERALIZED ZYGMUND CLASSES OF FUNCTIONS

We investigate the order of magnitude of the modulus of smoothness of a function f with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that f belongs to one of the generalized Zygmund classes Zyg(alpha, L) and Zyg(alpha, 1/L), where 0 <= alpha <= 2 and L = L(x) is a positive, nondecreasing, slowly varying function and such that L(x) -> infinity as x -> infinity. A continuous periodic function f with period 2 pi is said to belong to the class Zyg(alpha, L) if vertical bar f (x + h) - 2f (x) + f (x - h)vertical bar <= Ch(alpha) L (1/h) for all x is an element of T and h > 0, where the constant C does not depend on x and h; and the class Zyg(alpha, 1/L) is defined analogously. The above sufficient conditions are also necessary in case the Fourier coefficients of f are all nonnegative.